Tunneling time and Hartman effect in a ferromagnetic graphene superlattice

Using transfer-matrix and stationary phase methods, we study the tunneling time (group delay time) in a ferromagnetic monolayer graphene superlattice. The system we peruse consists of a sequence of rectangular barriers and wells, which can be realized by putting a series of electronic gates on the top of ferromagnetic graphene. The magnetization in the two ferromagnetic layers is aligned parallel. We find out that the tunneling time for normal incident is independent of spin state of electron as well as the barrier height and electron Fermi energy while for the oblique incident angles the tunneling time depends on the spin state of electron and has an oscillatory behavior. Also the effect of barrier width on tunneling time is also investigated and shown that, for normal incident, the Hartman effect disappears in a ferromagnetic graphene superlattice but it appears for oblique incident angles when the x component of the electron wave vector in the barrier is imaginary